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Algebra and trigonometry
Trigonometric identities and
Solving trigonometric equations
Try It
Use algebraic techniques to verify the identity:
cos
θ
1
+
sin
θ
=
1
−
sin
θ
cos
θ
.
(Hint: Multiply the numerator and denominator on the left side by
1
−
sin
θ
.
)
cos
θ
1
+
sin
θ
(
1
−
sin
θ
1
−
sin
θ
)
=
cos
θ
(
1
−
sin
θ
)
1
−
sin
2
θ
=
cos
θ
(
1
−
sin
θ
)
cos
2
θ
=
1
−
sin
θ
cos
θ
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Key equations
Pythagorean identities
cos
2
θ
+
sin
2
θ
=
1
1
+
cot
2
θ
=
csc
2
θ
1
+
tan
2
θ
=
sec
2
θ
Even-odd identities
tan
(
−
θ
)
=
−
tan
θ
cot
(
−
θ
)
=
−
cot
θ
sin
(
−
θ
)
=
−
sin
θ
csc
(
−
θ
)
=
−
csc
θ
cos
(
−
θ
)
=
cos
θ
sec
(
−
θ
)
=
sec
θ
Reciprocal identities
sin
θ
=
1
csc
θ
cos
θ
=
1
sec
θ
tan
θ
=
1
cot
θ
csc
θ
=
1
sin
θ
sec
θ
=
1
cos
θ
cot
θ
=
1
tan
θ
Quotient identities
tan
θ
=
sin
θ
cos
θ
cot
θ
=
cos
θ
sin
θ
Key concepts
There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
Graphing both sides of an identity will verify it. See
[link] .
Simplifying one side of the equation to equal the other side is another method for verifying an identity. See
[link] and
[link] .
The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. See
[link] .
We can create an identity and then verify it. See
[link] .
Verifying an identity may involve algebra with the fundamental identities. See
[link] and
[link] .
Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. See
[link] ,
[link] , and
[link] .
Section exercises
Verbal
We know
g
(
x
)
=
cos
x
is an even function, and
f
(
x
)
=
sin
x
and
h
(
x
)
=
tan
x
are odd functions. What about
G
(
x
)
=
cos
2
x
,
F
(
x
)
=
sin
2
x
, and
H
(
x
)
=
tan
2
x
?
Are they even, odd, or neither? Why?
All three functions,
F
,
G
, and
H
, are even.
This is because
F
(
−
x
)
=
sin
(
−
x
)
sin
(
−
x
)
=
(
−
sin
x
)
(
−
sin
x
)
=
sin
2
x
=
F
(
x
)
,
G
(
−
x
)
=
cos
(
−
x
)
cos
(
−
x
)
=
cos
x
cos
x
=
cos
2
x
=
G
(
x
)
and
H
(
−
x
)
=
tan
(
−
x
)
tan
(
−
x
)
=
(
−
tan
x
)
(
−
tan
x
)
=
tan
2
x
=
H
(
x
)
.
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Examine the graph of
f
(
x
)
=
sec
x
on the interval
[
−
π
,
π
]
.
How can we tell whether the function is even or odd by only observing the graph of
f
(
x
)
=
sec
x
?
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After examining the reciprocal identity for
sec
t
, explain why the function is undefined at certain points.
When
cos
t
=
0
, then
sec
t
=
1
0
, which is undefined.
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Algebraic
For the following exercises, use the fundamental identities to fully simplify the expression.
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
For the following exercises, verify the identity.
1
+
sin
2
x
cos
2
x
=
1
cos
2
x
+
sin
2
x
cos
2
x
=
1
+
2
tan
2
x
Answers will vary. Sample proof:
1
+
sin
2
x
cos
2
x
=
1
cos
2
x
+
sin
2
x
cos
2
x
=
sec
2
x
+
tan
2
x
=
tan
2
x
+
1
+
tan
2
x
=
1
+
2
tan
2
x
Got questions? Get instant answers now!
cos
2
x
−
tan
2
x
=
2
−
sin
2
x
−
sec
2
x
Answers will vary. Sample proof:
cos
2
x
−
tan
2
x
=
1
−
sin
2
x
−
(
sec
2
x
−
1
)
=
1
−
sin
2
x
−
sec
2
x
+
1
=
2
−
sin
2
x
−
sec
2
x
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Extensions
For the following exercises, prove or disprove the identity.
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.
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Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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